Let ℝ ⁿ (n ≥ 2) be n-dimensional Euclidian space and S ^n-1 be the unit sphere in ℝ ⁿ . Suppose that the function Ω ∈ L ¹(S ^n-1) satisfies the following cancelation condition

∫ S n - 1 Ω ( θ ) d σ ( θ ) = 0 ,

where dσ denotes the usual Lebesgue surface measure on the unit sphere S ^n-1.

Let L(log L)^α (S ^n-1) denotes the functions Ω defined on S ^n-1 satisfying the Zygmund condition: for α > 0,

∫ S n - 1 | Ω ( θ ) | ( log ( 2 + | Ω ( θ ) | ) ) α d σ ( θ ) < ∞ .

It is noted that for any q > 1, we have the proper inclusion relations hold:

L q ( S n - 1 ) ⊂ L ( log L ) α ( S n - 1 ) ⊂ L 1 ( S n - 1 ) , L ( log L ) β ( S n - 1 ) ⊂ L ( log L ) α ( S n - 1 ) if 0 < α < β .

For s ≥ 1, let Δ _s (ℝ⁺) denote the collection of measurable functions h on ℝ⁺ = {t ∈ ℝ : t > 0} satisfying

| | h | | Δ s ( ℝ + ) = sup j ∈ ℤ ∫ 2 j 2 j + 1 | h ( t ) | s d t ∕ t 1 ∕ s < ∞ ,

where ℤ denotes the set of integers. Also by usual modification, Δ_∞ (ℝ⁺) = L ^∞(ℝ⁺).

We note that Δ _s ⊂ Δ _t if s > t. We can always assume that h ∈ Δ₁.

A singular integral operator is defined in the following form:

S ( f ) ( x ) = p . v . ∫ ℝ n f ( x - y ) K ( y ) d y = lim ε → 0 ∫ | y | > ε f ( x - y ) K ( y ) d y ,

for an appropriate function f on ℝ ⁿ , where K (y) = |y|^-n h(|y|)Ω(y'), y' = |y|^-1 y.

It is well known that if Ω ∈ L log L(S ^n-1), h = 1, by the method of rotations, Calderón and Zygmund 1 proved that S extends to a bounded operator on L ^p for all p ∈ (1, ∞). In 2 , R. Fefferman first introduced the case of rough radial and proved that if h ∈ Δ_∞ (ℝ⁺) and Ω satisfy a Lipschitz condition of positive order on S ^n-1, then S is bounded on L ^p for 1 < p < ∞. Namazi 3 improved this result by replacing the Lipschitz condition by the condition that Ω ∈ L ^q (S ^n-1) for some q > 1. In 4 , Duoandikoetxea and Rubio de Francia developed some methods that can be used to study mapping properties of several kinds of operators in harmonic analysis, where they proved that S is bounded on L ^p for 1 < p < ∞ when h ∈ Δ₂(ℝ⁺) and Ω ∈ L ^q (S ^n-1). In 5 , Al-Salman and Pan proved that S is bounded on L^p for 1 < p < ∞ when h ∈ Δ_s(ℝ⁺)(s > 1) and Ω ∈ L log L(S ^n-1). Recently, using a method called Yano's extrapolation method 6 7 , Sato 8 proved that S extends to be an operator bounded on L ^p for 1 < p < ∞ where Ω ∈ L log L(S ^n-1) and the radial function h satisfying a rougher condition as a log type.

Define the function spaces

L a ( ℝ + ) = { h : h be measurable functions on ℝ  +  , L a ( h ) < ∞ } ,

where

L a ( h ) = sup j ∈ ℤ ∫ 2 j 2 j + 1 | h ( r ) | ( log ( 2 + | h ( r ) | ) ) a d r r .

And define the function space

N a ( ℝ + ) = { h : h be measurable functions on ℝ + , N a ( h ) < ∞ } ,

where

N a ( h ) = ∑ m ≥ 1 m a 2 m d m ( h ) ,

with d _m (h) = sup _k∈ℤ 2^-k |E(k, m)| and E(k, m) = {r ∈ (2 ^k , 2 ^k+1] : 2 ^m-1 < |h(r)| ≤ 2 ^m } for m ≥ 2, E(k, 1) = {r ∈ (2 ^k , 2 ^k+1] : |h(r)| ≤ 2}. Indeed, it is noted that for any a > 0, N a ( ℝ + ) ⊂ L a ( ℝ + ) and L a + b ( ℝ + ) ⊂ N a ( ℝ + ) for some b > 1.

Sato's main result is the following theorem:

Theorem A. 8 Suppose Ω is a function in L log L(S ^n-1) satisfying (1.1) and h ∈ N 1 ( ℝ + ) (or h ∈ L a ( ℝ + ) for some a > 2). Let S be as in (1.2). Then, there is a constant C such that

| | S ( f ) | | L p ( ℝ n ) ≤ C | | f | | L p ( ℝ n )

for all p ∈ (1, ∞).

For the one-parameter case, there are also several other papers. Especially, in 9 10 , weighted L ^p boundedness of singular integrals was discussed. The reader also can refer to 11 12 13 for more background materials.

In the article, we mainly consider the L ^p boundedness for the multiple singular integrals with rough kernels. Suppose that S ^d-1(d = n or m) is the unit sphere of ℝ ^d (d ≥ 2) equipped with the usual Lebesgue measure dσ. Let Ω ∈ L ¹(S ^n-1 × S ^m-1) satisfy the following double cancelation condition:

∫ S n - 1 Ω ( u , v ) d σ ( u ) = 0 and ∫ S m - 1 Ω ( u , v ) d σ ( v ) = 0 .

For α ≥ 1,

Δ α ( ℝ + × ℝ + ) = { h : h be measurable functions on ℝ + × ℝ + , | | h | | Δ α < ∞ } ,

where

| | h | | Δ α = sup k , j ∈ ℤ ∫ 2 k 2 k + 1 ∫ 2 j 2 j + 1 | h ( r , s ) | α d r d s r s 1 α .

The multiple singular integral on the product space ℝ ⁿ × ℝ ^m is defined by the following form:

T f ( x 1 , x 2 ) = p . v . ∫ ℝ n × ℝ m f ( x 1 - y 1 , x 2 - y 2 ) K ( y 1 , y 2 ) d y 1 d y 2

for an appropriate function f on ℝ ⁿ × ℝ ^m , where

K ( y 1 , y 2 ) = | y 1 | - n | y 2 | - m Ω ( y 1 ′ , y 2 ′ ) h ( | y 1 | , | y 2 | ) .

Let L(log L) ^α (S ^n-1 × S ^m-1) denote the class of the functions Ω defined on S ^n-1 × S ^m-1 satisfying the Zygmund condition: for α > 0,

∫ S n - 1 × S m - 1 | Ω ( θ , ω ) | ( log ( 2 + | Ω ( θ , ω ) | ) ) α d σ ( θ ) d σ ( ω ) < ∞ .

Historically, multiple singular integral was introduced by R. Fefferman and Stein's famous work on multiparameter harmonic analysis. Fefferman and Stein 14 proved that when h ≡ 1, T is bounded on L ^p (ℝ ⁿ × ℝ ^m ) for 1 < p < ∞ if Ω satisfy certain smooth conditions. Their method mainly relies on so-called square function method. Subsequently, in 15 , Duoandikoetxea used the method established in 4 and proved that T is bounded on L ^p (ℝ ⁿ × ℝ ^m ) for 1 < p < ∞ when Ω ∈ L ^q (S ^n-1 × S ^m-1) for some q > 1 and h ∈ Δ₂(ℝ⁺ × ℝ⁺). In 16 , Fan-Guo-Pan proved that T is bounded on L ^p (ℝ ⁿ × ℝ ^m ) for 1 < p < ∞ for the case when Ω belongs to certain block spaces that contain L ^q (S ^n-1 × S ^m-1) (for p = 2, it was proved by Jiang and Lu in 17 ) and h = 1. In 18 , Chen proved that T is bounded on L ^p (ℝ ⁿ × ℝ ^m ) for 1 < p < ∞ when Ω ∈ L(log L)² (S ^n-1 × S ^m-1) and h = 1 where he mainly relies on the method of rotation. In 19 , Al-Salman, Al-Qassem and Pan proved that T is bounded on L ^p (ℝ ⁿ × ℝ ^m ) for 1 < p < ∞ when Ω ∈ L(log L)²(S ^n-1 × S ^m-1) and h ∈ Δ _α for some α > 1, where their technique mostly based on refining the Duoandikoetxea-Rubia's Fourier transform estimates and Littlewood-paley theory. In the same paper, they also pointed out that for any ε > 0, there is a function Ω ∈ L(log L)^2-ε (S ^n-1 × S ^m-1) such that T may fail to be bounded on L ^p (ℝ ⁿ × ℝ ^m ).

The main purpose of this paper is to improve the above results, especially the rough product radial part. For this reason, we introduce several measurable function spaces defined on ℝ + × ℝ + : Δ ̃ α ( ℝ + × ℝ + ) , L α ( ℝ + × ℝ + ) and N α ( ℝ + × ℝ + ) for α > 0, where these spaces are equipped with the following "norms":

‖ h ‖ Δ ˜ α ( ℝ + × ℝ + ) = sup k ∈ ℤ ( ∫ 2 k 2 k + 1 sup j ∈ ℤ ∫ 2 j 2 j + 1 | h ( r , s ) | α d s s d r r ) 1 α + sup j ∈ ℤ ( ∫ 2 j 2 j + 1 sup k ∈ ℤ ∫ 2 k 2 k + 1 | h ( r , s ) | α d r r d s s ) 1 α ,               ‖ h ‖ ℒ α ( ℝ + × ℝ + ) = sup j , k ∈ ℤ ∫ 2 j 2 j + 1 ∫ 2 k 2 k + 1 | h ( r , s ) | ( log ( 2 + | h ( r , s ) | ) ) α d r d s r s ,                       ‖ h ‖ N α ( ℝ + × ℝ + ) = ∑ m ≥ 1 m a 2 m D m ( h ) ,

with D m ( h ) = sup k , j ∈ ℤ 2 - k 2 - j | E ( k , j , m ) | and E(k, j, m) = {(r, s) ∈ (2 ^k , 2 ^k+1] × (2 ^j , 2 ^j+1] : 2 ^m-1 < |h(r, s)| ≤ 2 ^m } for m ≥ 2, E(k, j, 1) = {(r, s) ∈ (2 ^k , 2 ^k+1] × (2 ^j , 2 ^j+1] : |h(r, s)| ≤ 2}.

Remark 1.1. Of course by the usual modification, Δ ∞ ( ℝ + × ℝ + ) = Δ ̃ ∞ ( ℝ + × ℝ + ) = L ∞ ( ℝ + × ℝ + ) . For simplicity, we let Δ α = Δ α ( ℝ + × ℝ + ) , Δ ̃ α = Δ ̃ α ( ℝ + × ℝ + ) , L α = L α ( ℝ + × ℝ + ) and N α = N α ( ℝ + × ℝ + ) . It is easy to check that (1) Δ ∞ ⊂ Δ ̃ α ⊂ Δ α ; (2) Δ ̃ α ⊂ Δ ̃ β if 1 ≤ β < α; (3) for any α > 0, N α ⊂ L α and L α + β ⊂ N α for any β > 1; (4) for any α > 1 and β > 0, Δ α ⊂ L β ⊂ Δ 1 .

Our main results are the following theorems:

Theorem 1.1. Suppose that Ω ∈ L(log L)²(S ^n-1 × S ^m-1) satisfying (1.3),

(1) if h ∈ N 2 or L α for some α > 3, then there is a constant C such that

| | T f | | L 2 ( ℝ n × ℝ m ) ≤ C | | f | | L 2 ( ℝ n × ℝ m ) .

(2) if h ∈ Δ ̃ α ( α ∈ ( 1 , 2 ] ) , then there is a constant C, which is independent of α, such that

| | T f | | L p ( ℝ n × ℝ m ) ≤ C 1 ( α - 1 ) 2 | | h | | Δ ̃ α | | f | | L p ( ℝ n × ℝ m ) ,

for p ∈ (1, ∞).

Remark 1.2. In 19 , it was proved that h ∈ Δ _α for some α > 1 and Ω ∈ L(log L)²(S ^n-1 × S ^m-1) are sufficient for L ^p boundedness for the multiple singular integral T. As for p = 2, Theorem 1.1 extended this result. For p ≠ 2, our condition h ∈ Δ ̃ α ( α ∈ ( 1 , 2 ] ) is strong. However, our result gives a sharp constant estimate, which gives the following corollary when the product radial part is separated (that is, h(r, s) = h ₁(r) ⋅ h ₂(s)).

Corollary 1.1. Suppose that Ω ∈ L(log L)²(S ^n-1 × S ^m-1) satisfying (1.3) and if h(r, s) = h ₁(r) ⋅ h₂(s), where h ₁ or h ₂ satisfies one of the following case:

(1) h ₁ ∈ Δ _α (ℝ⁺) ((α > 1)) and h 2 ∈ N 2 ( ℝ + ) (or h 2 ∈ L a ( ℝ + ) for some a > 3);

(2) h 1 ∈ N 2 ( ℝ + ) (or h 1 ∈ L a ( ℝ + ) fsor some a > 3) and h ₂ ∈ Δ _s (ℝ⁺) ((s > 1)),

then there is a constant C such that

| | T f | | L p ( ℝ n × ℝ m ) ≤ C | | f | | L p ( ℝ n × ℝ m ) ,

for p ∈ (1, ∞).

Our proof of the above theorem is based on the argument of Sato 8 , which mainly relied on Yano's extrapolation method. The following theorem is the key step to prove Theorem 1.1.

Theorem 1.2. Suppose that Ω ∈ L ^q (S ^n-1 × S ^m-1)(q ∈ (1, 2]) satisfying (1.3).

(1) If h ∈ Δ _α (α ∈ (1, 2]), then there exists a constant C, which is independent of q, α, Ω, h, such that

| | T f | | L 2 ( ℝ n × ℝ m ) ≤ C 1 ( α - 1 ) 2 1 ( q - 1 ) 2 | | h | | Δ α | | Ω | | L p ( S n - 1 × S m - 1 ) | | f | | L 2 ( ℝ n × ℝ m ) .

(2) If h ∈ Δ ̃ α ( α ∈ ( 1 , 2 ] ) , then there exists a constant C, which is independent of q, α, Ω, h, such that

| | T f | | L p ( ℝ n × ℝ m ) ≤ C 1 ( α - 1 ) 2 1 ( q - 1 ) 2 | | h | | Δ ̃ α | | Ω | | L q ( S n - 1 × S m - 1 ) | | f | | L p ( ℝ n × ℝ m ) ,

for p ∈ (1, ∞).

Remark 1.3. Corollary 1 in 15 asserted that if h ∈ Δ₂ and Ω ∈ L ^q (q > 1)(S ^n-1 × S ^m-1), then T is bounded in L ^p (ℝ ⁿ × ℝ ^m ) for p > 1. After a careful check of its proof, we find that the condition h ∈ Δ₂ is not sufficient for p ≠ 2 since the two partial maximal functions are taken supremum both j and k, it seems that if h ∈ Δ₂, the partial maximal function is not pointwise controlled by the one-parameter maximal function case (line 10-13, 15 ). If we substitute h ∈ Δ₂ with h ∈ Δ ̃ 2 , Corollary 1 in 15 is corrected. This is why we introduce the space Δ ̃ α . Of course, we remark that our result is mainly influenced by the idea and the technique established in 15 : Littlewood-Paley theory for product theory, Fourier transform estimates, etc.

Remark 1.4. The maximal multiple singular integral is defined as

T * f ( x 1 , x 2 ) = sup ε 1 > 0 , ε 2 > 0 ∬ | y 1 | ≥ ε 1 , | y 2 | ≥ ε 2 f ( x 1 - y 1 , x 2 - y 2 ) K ( y 1 , y 2 ) d y 1 d y 2 ,

where K is as in (1.5). By the estimates we have established and Yano's extrapolation method, combining with 20 , we have the same result for the maximal multiple singular integral as in 19 :

Theorem 1.3. Suppose that Ω ∈ L(log L)²(S ^n-1 × S ^m-1) satisfies (1.3) and h ∈ Δ_∞, then there exists a constant C, such that

| | T * f | | L p ( ℝ n × ℝ m ) ≤ C | | h | | Δ ∞ | | f | | L p ( ℝ n × ℝ m ) ,

for p ∈ (1, ∞).

We leave the proof to the interested reader. But we do not know whether h can be extended to more general case like Δ ̃ α ( 1 < α < ∞ ) .

This paper is organized as the following. In Section 2, we give the proof of Theorem 1.2. In Section 3, we give the proof of Theorem 1.1 and Corollary 1.1. Throughout this paper, the letter C will stand for a constant that may vary at each occurrence but that is independent of the essential variables and p' be the conjugation of p satisfying 1 p + 1 p ′ = 1 .

Let Ω, h be as in Theorem 1.2. We let ρ ≥ 2, define

E k , j = { ( y 1 , y 2 ) ∈ ℝ n + m : ρ k < | y 1 | ≤ ρ k + 1 , ρ j < | y 2 | ≤ ρ j + 1 }

and measures σ _k,j by

σ k , j * f ( x 1 , x 2 ) = ∬ E k , j K ( y 1 , y 2 ) f ( x 1 - y 1 , x 2 - y 2 ) d y 1 d y 2 .

So

T f = ∑ k , j σ k , j * f .

Define σ* by σ* f(x) = sup _k,j ||σ _k,j | * f(x)|, where |σ _k,j | denotes the total variation. Let μ _k,j = |σ _k,j | and define μ* by μ* f(x) = sup _k,j |μ _k,j * f(x)|. Let θ ∈ (0, 1), δ(p) = |1/p - 1/p'|, we have the following two lemmas.

Lemma 2.1. For p > 1 + θ, suppose that Ω ∈ L ^q (S ^n-1 × S ^m-1)(q ∈ (1,2]) satisfying (1.3) and h ∈ Δ ̃ α ( α ∈ ( 1 , 2 ] ) , we have

| | μ * f | | L p ( ℝ n × ℝ m ) ≤ C log 2 ρ | | Ω | | L q ( S n - 1 × S m - 1 ) | | h | | Δ ̃ α 1 - 2 - θ 2 - 2 . 2 ∕ p | | f | | L p ( ℝ n × ℝ m ) ,

where the constant C is independent of q, α, Ω, h.

Lemma 2.2. (1). Suppose that Ω ∈ L ^q (S ^n-1 × S ^m-1)(q ∈ (1,2]) satisfying (1.3) and h ∈ Δ _α (α ∈ (1,2]),

| | T f | | L 2 ( ℝ n × ℝ m ) ≤ C log 2 ρ | | Ω | | L q ( S n - 1 × S m - 1 ) | | h | | Δ α | | f | | L 2 ( ℝ n × ℝ m ) ,

(2). For p ∈ (1 + θ, (1 + θ)/θ), suppose that Ω ∈ L ^q (S ^n-1 × S ^m-1)(q ∈ (1, 2]) satisfying (1.3) and h ∈ Δ ̃ α ( α ∈ ( 1 , 2 ] ) , we have

| | T f | | L p ( ℝ n × ℝ m ) ≤ C log 2 ρ | | Ω | | L q ( S n - 1 × S m - 1 ) | | h | | Δ ̃ α 1 - 2 - θ 2 - 2 ( 1 + δ ( p ) ) | | f | | L p ( ℝ n × ℝ m ) ,

where the constant C is independent of q, α, Ω, h.

If Lemma 2.2 is proved, since θ ∈ (0, 1) is arbitrary and we choose ρ = 2 ^q'α' , then Theorem 1.2 is an immediate consequence of Lemma 2.2 immediately.

Now, we prove part (1) of Lemma 2.2. For simplicity, we let A = log 2 ρ | | Ω | | L q ( S n - 1 × S m - 1 ) | | h | | Δ α . Firstly, we have the following estimates for the measures σ _k,j :

| | σ k , j | | ≤ c 1 A

| σ ^ k , j ( ξ 1 , ξ 2 ) ≤ c 2 A | ρ k ξ 1 | ± 1 2 q ′ α ′ | ρ j ξ 2 | ± 1 2 q ′ α ′

for some constants c _i . The equation (2.4) is the consequence of the following result:

| | σ k , j | | = | σ k , j | ( ℝ n × ℝ m ) ≤ ∫ ρ k ρ k + 1 ∫ ρ j ρ j + 1 ∫ S n - 1 × S m - 1 | Ω ( u , v ) | | h ( r , s ) | d σ ( u ) d σ ( v ) d r d s r s ≤ C log 2 ρ | | Ω | | L 1 ( S n - 1 × S m - 1 ) | | h | | Δ 1 .

Now, we turn to prove (2.5), note

σ ^ k , j ( ξ 1 , ξ 2 ) = ∫ ρ k ρ k + 1 ∫ ρ j ρ j + 1 ∫ S n - 1 × S m - 1 Ω ( u , v ) h ( r , s ) e - 2 π i ( ξ 1 ⋅ r u + ξ 2 ⋅ s v ) d σ ( u ) d σ ( v ) d r d s r s

and we define

F ( r , s , ξ 1 , ξ 2 ) = ∫ S n - 1 × S m - 1 Ω ( u , v ) e - 2 π i ( ξ 1 ⋅ r u + ξ 2 ⋅ s v ) d σ ( u ) d σ ( v ) .

Then, by Hölder's inequality,

| σ ^ k , j ( ξ 1 , ξ 2 ) | = ∫ ρ k ρ k + 1 ∫ ρ j ρ j + 1 F ( r , s , ξ 1 , ξ 2 ) h ( r , s ) d r d s r s ≤ ∫ ρ k ρ k + 1 ∫ ρ j ρ j + 1 | h ( r , s ) | α d r d s r s 1 α ∫ ρ k ρ k + 1 ∫ ρ j ρ j + 1 | F ( r , s , ξ 1 , ξ 2 ) | α ′ d r d s r s 1 α ′ ≤ ∫ ρ k ρ k + 1 ∫ ρ j ρ j + 1 | h ( r , s ) | α d r d s r s 1 α | | Ω | | L 1 ( S n - 1 × S m - 1 ) α ′ - 2 α ′ ∫ ρ k ρ k + 1 ∫ ρ j ρ j + 1 | F ( r , s , ξ 1 , ξ 2 ) | 2 d r d s r s 1 α ′

while here

∫ ρ k ρ k + 1 ∫ ρ j ρ j + 1 | F ( r , s , ξ 1 , ξ 2 ) | 2 d r d s r s = ∫ ρ k ρ k + 1 ∫ ρ j ρ j + 1 ∬ ( S n - 1 × S m - 1 ) 2 Ω ( u , v ) Ω ( u ′ , v ′ ) ¯ e - 2 π i ( ξ 1 ⋅ r ( u - u ′ ) + ξ 2 ⋅ s ( v - v ′ ) ) d σ ( u ′ ) d σ ( v ) d σ ( v ′ ) d r d s r s ≤ ∬ ( S n - 1 × S m - 1 ) 2 Ω ( u , v ) Ω ( u ′ , v ′ ) ¯ ∫ ρ k ρ k + 1 ∫ ρ j ρ j + 1 e - 2 π i ( ξ 1 ⋅ r ( u - u ′ ) + ξ 2 ⋅ s ( v - v ′ ) ) d r d s r s d σ ( u ) d σ ( u ′ ) d σ ( v ) d σ ( v ′ ) ≤ C log 2 ρ | | Ω | | L q ( S n - 1 × S m - 1 ) 2 | ρ k ξ 1 | - ε | ρ j ξ 2 | - ε ⋅ ∫ S n - 1 × S n - 1 d σ ( u ) d σ ( u ′ ) | ξ ′ 1 ⋅ ( u - u ′ ) | ε q ′ 1 q ′ ∫ S m - 1 × S m - 1 d σ ( v ) d σ ( v ′ ) | ξ ′ 2 ⋅ ( v - v ′ ) | ε q ′ 1 q ′ .

When εq' < 1 (indeed we set ε = 1 2 q ′ ), the integrals ∫ s n - 1 × S n - 1 d σ ( u ) d σ ( u ′ ) | ξ ′ 1 ⋅ ( u - u ′ ) | ε q ′ 1 α ′ q ′ and ∫ S m - 1 × S m - 1 d σ ( v ) d σ ( v ′ ) | ξ ′ 2 ⋅ ( v - v ′ ) | ε q ′ 1 α ′ q ′ are finite and independent of q and α. So we have

| σ ^ k , j ( ξ 1 , ξ 2 ) | ≤ C log 2 ρ | | h | | Δ α | | Ω | | L q ( S n - 1 × S m - 1 ) | ρ k ξ 1 | - 1 2 q ′ α ′ | ρ j ξ 2 | - 1 2 q ′ α ′ .

Since Ω satisfies the condition (1.3), we have σ ^ k , j ( 0 , ξ 2 ) = 0 and then | σ ^ k , j ( ξ 1 , ξ 2 ) | equals to

| σ ^ k , j ( ξ 1 , ξ 2 ) - σ ^ k , j ( 0 , ξ 2 ) | = ∫ ρ k ρ k + 1 ∫ ρ j ρ j + 1 ∫ S n - 1 × S m - 1 Ω ( u , v ) h ( r , s ) [ e - 2 π i ξ 1 ⋅ r u - 1 ] e - 2 π i ξ 2 ⋅ s v d σ ( u ) d σ ( v ) d r d s r s ≤ ∫ ρ k ρ k + 1 ∫ S n - 1 ∫ ρ j ρ j + 1 ∫ S m - 1 Ω ( u , v ) h ( r , s ) e - 2 π i ξ 2 ⋅ s v d v d s s | e - 2 π i ξ 1 ⋅ r u - 1 | d u d r r ≤ ∫ ρ k ρ k + 1 ∫ S n - 1 ∫ ρ j ρ j + 1 ∫ S m - 1 Ω ( u , v ) h ( r , s ) e - 2 π i ξ 2 ⋅ s v d v d s s min ( { 2 , r | ξ 1 | } ) d u d r r ≤ C log 2 ρ | | h | | Δ α | | Ω | | L q ( S n - 1 × S m - 1 ) | ρ k ξ 1 | 1 2 q ′ α ′ | ρ j ξ 2 | - 1 2 q ′ α ′ .

The same way as above, we have | σ ^ k , j ( ξ 1 , ξ 2 ) | equals to

| σ ^ k , j ( ξ 1 , ξ 2 ) - σ ^ k , j ( ξ 1 , 0 ) | ≤ C log 2 ρ | | h | | Δ α | | Ω | | L q ( S n - 1 × S m - 1 ) | ρ k ξ 1 | - 1 2 q ′ α ′ | ρ j ξ 2 | 1 2 q ′ α ′ .

Also we have | σ ^ k , j ( ξ 1 , ξ 2 ) | equals to

| σ ^ k , j ( ξ 1 , ξ 2 ) - σ ^ k , j ( ξ 1 , 0 ) - σ ^ k , j ( 0 , ξ 2 ) + σ ^ k , j ( 0 , 0 ) | ≤ C log 2 ρ | | h | | Δ α | | Ω | | L q ( S n - 1 × S m - 1 ) | ρ k ξ 1 | 1 2 q ′ α ′ | ρ j ξ 2 | 1 2 q ′ α ′ .

Consequently, the inequality (2.5) is just the combination of (2.6), (2.7),(2.8) and (2.9).

Let ψ 1 ∈ S ( ℝ n ) , ψ 2 ∈ S ( ℝ m ) , such that

s u p p ( ψ i ( ξ i ) ) ⊂ { 1 ρ ≤ | ξ i | < ρ } , i = 1 , 2 , 0 ≤ ψ i ( ξ i ) ≤ 1 , i = 1 , 2 ,

and

∑ k = - ∞ ∞ | ( ψ 1 ) ( ρ k ξ 1 ) | 2 = ∑ j = - ∞ ∞ | ( ψ 2 ) ( ρ j ξ 2 ) | 2 = 1 .